The 2CNF Boolean Formula Satisfiability Problem and the Linear Space Hypothesis

We aim at investigating the solvability/insolvability of nondeterministic logarithmic-space (NL) decision, search, and optimization problems parameterized by size parameters using simultaneously polynomial time and sub-linear space on multi-tape deterministic Turing machines. We are particularly focused on a special NL-complete problem, 2SAT---the 2CNF Boolean formula satisfiability problem---parameterized by the number of Boolean variables. It is shown that 2SAT with $n$ variables and $m$ clauses can be solved simultaneously polynomial time and $(n/2^{c\sqrt{\log{n}}})\, polylog(m+n)$ space for an absolute constant $c>0$. This fact inspires us to propose a new, practical working hypothesis, called the linear space hypothesis (LSH), which states that 2SAT$_3$---a restricted variant of 2SAT in which each variable of a given 2CNF formula appears at most 3 times in the form of literals---cannot be solved simultaneously in polynomial time using strictly "sub-linear" (i.e., $m(x)^{\varepsilon}\, polylog(|x|)$ for a certain constant $\varepsilon\in(0,1)$) space on all instances $x$. An immediate consequence of this working hypothesis is $\mathrm{L}\neq\mathrm{NL}$. Moreover, we use our hypothesis as a plausible basis to lead to the insolvability of various NL search problems as well as the nonapproximability of NL optimization problems. For our investigation, since standard logarithmic-space reductions may no longer preserve polynomial-time sub-linear-space complexity, we need to introduce a new, practical notion of "short reduction." It turns out that, parameterized with the number of variables, $\overline{\mathrm{2SAT}_3}$ is complete for a syntactically restricted version of NL, called Syntactic NL$_{\omega}$, under such short reductions. This fact supports the legitimacy of our working hypothesis.Comment: (A4, 10pt, 25 pages) This current article extends and corrects its preliminary report in the Proc. of the 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), August 21-25, 2017, Aalborg, Denmark, Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik 2017, vol. 83, pp. 62:1-62:14, 201

Survey on the Tukey theory of ultrafilters

This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell's Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page

Pi01 encodability and omniscient reductions

A set of integers $A$ is computably encodable if every infinite set of integers has an infinite subset computing $A$. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this paper, we extend this notion of computable encodability to subsets of the Baire space and we characterize the $\Pi^0_1$ encodable compact sets as those who admit a non-empty $\Sigma^1_1$ subset. Thanks to this equivalence, we prove that weak weak K\"onig's lemma is not strongly computably reducible to Ramsey's theorem. This answers a question of Hirschfeldt and Jockusch.Comment: 9 page

Cardinal characteristics and countable Borel equivalence relations

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$. We analyze some of the basic behavior of these properties, showing for instance that the property corresponding to the splitting number $\mathfrak s$ coincides with smoothness. We then settle many of the implication relationships between the properties; these relationships turn out to be closely related to (but not the same as) the Borel Tukey ordering on cardinal characteristics

Computational Complexity for Physicists

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm